# Posterior consistency for scale-mixture shrinkage priors in low dimension?

Consider the model 

$$y_n=X_n\beta_n+\epsilon_n$$ $$\beta_i|\sigma^2,v_i \sim \mathcal{N}(0,\sigma^2 v_i), i=1,\ldots,p$$ $$v_i \sim \beta^\prime(a,b)$$ $$\sigma^2 \sim \mathcal{IG}(c,d)$$

where $$\beta^\prime$$ is the beta prime distribution and $$\mathcal{IG}$$ the inverse gamma distribution.

In , the authors prove posterior consistency in the high-dimensionality regimes were $$p\rightarrow \infty$$ as $$n\rightarrow \infty$$. Is there a way to show posterior consistency for fixed/small $$p$$ as $$n\rightarrow \infty$$?

 Bai, R. and Ghosh, M., 2021. On the beta prime prior for scale parameters in high-dimensional bayesian regression models. Statistica Sinica, 31(2), pp.843-865.

• @SextusEmpiricus, in arxiv.org/pdf/1807.06539.pdf the authors set $f$ to be a beta prime prior and prove posterior consistency in high dimension. My question is there a similar prove for fixed/small $p$ with the same prior and or different priors?
– MrDi
Mar 6 at 21:54
• @SextusEmpiricus, I think it is very familiar to anyone following the latest research of normal-scale mixture models and posterior consistency.
– MrDi
Mar 6 at 22:11
• @Ben, I have edited my question.
– MrDi
Mar 8 at 6:59
• @D.W., $f$ is the beta prime distribution.
– MrDi
Mar 8 at 7:00
• @SextusEmpiricus $\beta$ are the coefficients.
– MrDi
Mar 8 at 7:02

Based on your reference I believe that you are estimating the vector $$\boldsymbol{\beta}$$ of size $$p_n$$ with a posterior distribution based on the observation of the vector $$\mathbf{Y}$$ of size $$n$$ in the model $$\mathbf{Y} = \mathbf{X} \boldsymbol{\beta} + \boldsymbol{\epsilon}$$ where $$\mathbf{X}$$ is a fixed regression matrix $$n \times p_n$$ matrix. The components $$\beta_i$$ from the vector $$\boldsymbol{\beta}$$ have a prior distribution that follows the model that you described.

If you have a fixed $$p$$ then consistency seems guaranteed by a Bayesian law of large numbers, or consistency of likelihood/posterior for an i.i.d sample (When do posteriors converge to a point mass?). Or maybe I am missing something?

Say we use some fixed $$p$$ while letting $$n$$ increase, then the Bayesian posterior density $$f(\boldsymbol{\beta}|\mathbf{Y})$$ will concentrate near the true $$\boldsymbol{\beta}$$ (if the prior is not zero).

The same is true when $$p$$ is not fixed, but still with a finite upper bound. We can consider all $$p$$'s below the bound together while letting $$n \to \infty$$ and if they are all individually consistent, then we will also have consistency when we change $$p$$ while letting $$n$$ increase.

• I don't think this answers OP question. Mar 8 at 7:15
• @Isaac I answered the question "Is there a way to show posterior consistency for fixed/small $p$" by explaining a way that it can be shown. Could you tell what is wrong with it? Mar 8 at 7:40
• @SextusEmpiricus, could you elaborate more about this? How will the theorem in  change with fixed/small $p$?
– MrDi
Mar 8 at 9:09
• @MrDi for a fixed $p$ one can use the law of large numbers. When $p$ is not fixed then that approach becomes difficult because while we can have that for a larger $p$ the 'error' in the posterior may become larger. Then consistency is not obtained, if the increase in error due to increase in $p$ is faster than the decrease in error due to increase in $n$. Mar 8 at 9:20
• @SextusEmpiricus, could you please reference some source where the law of large numbers is used to prove posterior consistency for the normal-scale mixture model showen the question?
– MrDi
Mar 8 at 9:45